1.3: Proportions and Rates - Mathematics

If you wanted to power the city of Seattle using wind power, how many windmills would you need to install? Questions like these can be answered using rates and proportions.

Rates

A rate is the ratio (fraction) of two quantities.

A unit rate is a rate with a denominator of one.

Example 12

Your car can drive 300 miles on a tank of 15 gallons. Express this as a rate.

Solution

Expressed as a rate, (frac{300 ext { miles }}{15 ext { gallons }}). We can divide to find a unit rate: (frac{20 ext { miles }}{ ext { 1gallon }}), which we could also write as (20 frac{ ext { miles }}{ ext { gallon }}), or just 20 miles per gallon.

Proportion Equation

A proportion equation is an equation showing the equivalence of two rates or ratios.

Example 13

Solve the proportion (frac{5}{3} = frac{x}{6}) for the unknown value (x).

Solution

This proportion is asking us to find a fraction with denominator 6 that is equivalent to the fraction (frac{5}{3}). We can solve this by multiplying both sides of the equation by 6, giving (x=frac{5}{3} cdot 6=10).

Example 14

A map scale indicates that (frac{1}{2}) inch on the map corresponds with 3 real miles. How many miles apart are two cities that are (2 frac{1}{4}) inches apart on the map?

Solution

We can set up a proportion by setting equal two (frac{ ext { map inches }}{ ext { real miles }}) rates, and introducing a variable, (x), to represent the unknown quantity – the mile distance between the cities.

( egin{array} {ll} {frac{frac{1}{2} ext { map inch }}{3 ext { miles }}=frac{2 frac{1}{4} ext { map inches }}{x ext { miles }}} &{ ext{Multiply both sides by }x ext{ and rewriting the mixed number}} {frac{frac{1}{2}}{3} cdot x=frac{9}{4}} & { ext{Multiply both sides by 3}} {frac{1}{2} x=frac{27}{4}} &{ ext{Multiply both sides by 2 (or divide by }frac{1}{2})} {x=frac{27}{2}=13 frac{1}{2} ext { miles }} &{ } end{array} )

Many proportion problems can also be solved using dimensional analysis, the process of multiplying a quantity by rates to change the units.

Example 15

Your car can drive 300 miles on a tank of 15 gallons. How far can it drive on 40 gallons?

Solution

We could certainly answer this question using a proportion: ($frac{300 ext { miles }}{15 ext { gallons }}=frac{x ext { miles }}{40 ext { gallons }}$).

However, we earlier found that 300 miles on 15 gallons gives a rate of 20 miles per gallon. If we multiply the given 40 gallon quantity by this rate, the gallons unit “cancels” and we’re left with a number of miles:

(40 ext{gallons} cdot frac{20 ext { miles }}{ ext { gallon }}=frac{40 ext { gallons }}{1} cdot frac{20 ext { miles }}{ ext { gallon }}=800 ext{ miles})

Notice if instead we were asked “how many gallons are needed to drive 50 miles?” we could answer this question by inverting the 20 mile per gallon rate so that the miles unit cancels and we’re left with gallons:

(50 ext{miles} cdot frac{ ext { lgallon }}{20 ext { miles }}=frac{50 ext { miles }}{1} cdot frac{1 ext { gallon }}{20 ext { miles }}=frac{50 ext { gallons }}{20}=2.5 ext{ gallons})

Dimensional analysis can also be used to do unit conversions. Here are some unit conversions for reference.

Unit Conversions

Length

( egin{array} {ll} {1 ext { foot (ft)}=12 ext { inches (in)}} & {1 ext{ yard (yd) }=3 ext{ feet (ft)}} {1 ext{ mile }=5,280 ext{ feet}} & { } {1000 ext{ millimeters }{mm}=1 ext{ meter (m)}} & {100 ext{ centimeters (cm)}=1 ext{ meter}} {1000 ext{ meters (m)}=1 ext{ kilometer (km)}} &{2.54 ext{ centimeters (cm) }= 1 ext{ inch}} end{array} )

Weight and Mass

( egin{array} {ll} {1 ext{ pound (lb) }= 16 ext{ ounces (oz)}} & {1 ext{ ton }= 2000 ext{ pounds}} {1000 ext{ milligrams (mg) }= 1 ext{ gram (g)}} & {1000 ext{ grams }= 1 ext{ kilogram (kg)}} {1 ext{ kilogram }= 2.2 ext{ pounds (on earth)}} & { } end{array} )

Capacity

( egin{array} {ll} {1 ext{ cup }= 8 ext{ fluid ounces (fl oz)}^*} & {1 ext{ pint }= 2 ext{ cups}} {1 ext{ quart }= 2 ext{ pints }= 4 ext{ cups}} & {1 ext{ gallon }= 4 ext{ quarts }= 16 ext{ cups}} {1000 ext{ milliliters (ml) }= 1 ext{ liter (L)}} & { } end{array} )

*Fluid ounces are a capacity measurement for liquids. 1 fluid ounce ≈ 1 ounce (weight) for water only.

Example 16

A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds?

Solution

To answer this question, we need to convert 20 seconds into feet. If we know the speed of the bicycle in feet per second, this question would be simpler. Since we don’t, we will need to do additional unit conversions. We will need to know that 5280 ft = 1 mile. We might start by converting the 20 seconds into hours:

( egin{array} {ll} {20 ext{ seconds } cdot frac{1 ext { minute }}{60 ext { seconds }} cdot frac{1 ext { hour }}{60 ext { minutes }}=frac{1}{180} ext{ hour}} & { ext{Now we can multiply by the }15 ext{ miles/hr}} {frac{1}{180} ext { hour } cdot frac{15 ext { miles }}{ ext { Ihour }}=frac{1}{12} ext { mile }} & { ext{Now we can convert to feet}} {frac{1}{12} ext { mile } cdot frac{5280 ext { feet }}{1 ext { mile }}=440 ext { feet}} & { } end{array} )

We could have also done this entire calculation in one long set of products:

(20 ext{ seconds }cdot frac{1 ext { minute }}{60 ext { seconds }} cdot frac{1 ext { hour }}{60 ext { minutes }} cdot frac{15 ext { miles }}{1 ext { hour }} cdot frac{5280 ext { feet }}{1 ext { mile }}=440 ext{ feet})

Try it Now 4

A 1000 foot spool of bare 12-gauge copper wire weighs 19.8 pounds. How much will 18 inches of the wire weigh, in ounces?

(18 ext { inches } cdot frac{1 ext { foot }}{12 ext { inches }} cdot frac{19.8 ext { pounds }}{1000 ext { feet }} cdot frac{16 ext { ounces }}{1 ext { pound }} approx 0.475 ext { ounces })

Notice that with the miles per gallon example, if we double the miles driven, we double the gas used. Likewise, with the map distance example, if the map distance doubles, the real-life distance doubles. This is a key feature of proportional relationships, and one we must confirm before assuming two things are related proportionally.

Example 17

Suppose you’re tiling the floor of a 10 ft by 10 ft room, and find that 100 tiles will be needed. How many tiles will be needed to tile the floor of a 20 ft by 20 ft room?

Solution

In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, 400 tiles will be needed. We could find this using a proportion based on the areas of the rooms:

(frac{100 ext { tiles }}{100 mathrm{ft}^{2}}=frac{n ext { tiles }}{400 mathrm{ft}^{2}})

Other quantities just don’t scale proportionally at all.

Example 18

Suppose a small company spends $1000 on an advertising campaign, and gains 100 new customers from it. How many new customers should they expect if they spend$10,000?

Solution

While it is tempting to say that they will gain 1000 new customers, it is likely that additional advertising will be less effective than the initial advertising. For example, if the company is a hot tub store, there are likely only a fixed number of people interested in buying a hot tub, so there might not even be 1000 people in the town who would be potential customers.

Sometimes when working with rates, proportions, and percents, the process can be made more challenging by the magnitude of the numbers involved. Sometimes, large numbers are just difficult to comprehend.

Example 19

Compare the 2010 U.S. military budget of $683.7 billion to other quantities. Here we have a very large number, about$683,700,000,000 written out. Of course, imagining a billion dollars is very difficult, so it can help to compare it to other quantities.

If that amount of money was used to pay the salaries of the 1.4 million Walmart employees in the U.S., each would earn over $488,000. There are about 300 million people in the U.S. The military budget is about$2,200 per person.

If you were to put $683.7 billion in$100 bills, and count out 1 per second, it would take 216 years to finish counting it.

Example 20

Compare the electricity consumption per capita in China to the rate in Japan.

To address this question, we will first need data. From the CIA[1] website we can find the electricity consumption in 2011 for China was 4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the consumption for Japan was 859,700,000,000, or 859.7 billion KWH. To find the rate per capita (per person), we will also need the population of the two countries. From the World Bank[2], we can find the population of China is 1,344,130,000, or 1.344 billion, and the population of Japan is 127,817,277, or 127.8 million.

Solution

Computing the consumption per capita for each country:

China: (frac{4,693,000,000,000 mathrm{KWH}}{1,344,130,000 ext { people }} approx 3491.5) KWH per person

Japan: (frac{859,700,000,000 mathrm{KWH}}{127,817,277 ext { people }} approx 6726) KWH per person

While China uses more than 5 times the electricity of Japan overall, because the population of Japan is so much smaller, it turns out Japan uses almost twice the electricity per person compared to China.

[1] www.cia.gov/library/publicat.../2042rank.html

[2] http://data.worldbank.org/indicator/SP.POP.TOT

Significant Changes Due to the CCSSM introduction of ratio and rates in 6 th grade, the content of this unit has been combined into three investigation rather than 4.

CMP2&rsquos Comparing and Scaling has been altered in response to several forces, some internal and some external. There is a cut back (now only 3 investigations) builds on foundational work in 6 th grade Units (Comparing Bits & Pieces, Decimal Ops, and Variables & Patterns) and in 7th Grade Unit (Stretching & Shrinking). The internal motivation for change was a desire to more strongly connect to, and to more completely develop, the ideas and strategies initiated in CMP3 Stretching and Shrinking. In similar figures, internal ratios, such as length: width, and external ratios, such as side length X: corresponding side length Y, are constant. When students find a missing length in Stretching and Shrinking, they are solving a proportion they may use a scale factor rather than set up a proportion, but the proportional reasoning is the same. By the end of CMP3 Stretching and Shrinking students will have a method of scaling a ratio up or down to find a missing quantity or by using equivalent ratios, which is an important strategy for solving any proportion. Learning to choose the scale factor efficiently, in similarity and other situations, is a refinement of this strategy. They also use equivalent ratios to find missing length.

In CMP3 Comparing and Scaling, students use proportional reasoning in contexts other than geometric contexts, and develop additional strategies for solving proportions, including efficient scaling and common denominators. They will see that rate tables are a variation on a scaling strategy, and that unit rates are particularly useful. In the example below, number of units of length of base of triangle: unit length of height of triangle = 1.5: 1. Using this unit rate is another way to solve the proportion 3:2 = x: 11.

The external motivations for changing Comparing and Scaling were caused by the requirement, in CCSSM, of the study of ratio in Grade 6, and the addition of constant of proportionality to the content in grade 7. Because students are expected to have some understanding of ratio when they enter grade 7, less time is needed to introduce ratios and rates. Therefore, more time can be spent on developing proportional reasoning and connections among rates, ratios, rate tables and proportions. In addition, since the constant of proportionality implies connections among tables, graphs and equations of proportional relationships, more can be done with these representations. This in turn provides connections to the next unit, Moving Straight Ahead, where students will see that equations of proportional relationships are one kind of linear equation, and the constant of proportionality is, in fact, the slope of the line, or the rate of increase in the table.

Direct and Inverse Proportion

Direct Proportion

• The table below shows the cost of various numbers of cups at sh. 20 per cup.
 No. of cups 1 2 3 4 5 Cost (sh.) 20 40 60 80 100
• The ratio of the numbers of cups in the fourth column to the number of cups in the second column is 4:2=2:1 .
• The ratio of the corresponding costs is 80:40=2:1 . By considering the ratio of costs in any two columns and the corresponding ratio and the number of cups, you should notice that they are always the same.
• If two quantities are such that when the one increases (decreases) in particular ratio, the other one also increases (decreases) in the ratio, the two quantities are said to be in direct proportion

A car travels 40km on 5 litres of petrol. How far does it travel on 12 litres of petrol?

Petrol is increased in the ratio 12: 5
Distance= 40 x 12 /5 km

A train takes 3 hours to travel between two stations at an average speed of 40km per hour. A t what average speed would it need to travel to cover the same distance in 2hours?

Time is decreased in the ratio 2:3 Speed must be increased in the ratio 3:2 average speed is 40 x 3 /2 km = 60 km/h

Ten men working six hours a day take 12 days to complete a job. How long will it take eight men working 12 hours a day to complete the same job?

Number of men decreases in the ratio 8:10
Therefore, the number of days taken increases in the ratio 10:8.
Number of hours increased in the ratio 12:6.
Therefore, number of days decreases in the ratio 6:1 2.
Number of day s taken =12 x 10 /8 x 6 /12

Definitions

Did you notice I didn't give definitions of the terms ratio and proportion? Well, I didn't want to confuse. Sometimes you don't have to learn the exact definitions up front, but you can start by learning to solve word problems &mdash even real-life problems.

A RATIO is two "things" (numbers or quantities) compared to each other. For example, "3 dollars per gallon" is a ratio, and "40 miles per 1 hour" is another. Here are some more: 15 girls versus 14 boys, 569 words in 2 minutes, 23 green balls to 41 blue balls. Your math book might say it is a comparison of two numbers or quantities.

A related term, RATE, is defined as a ratio where the two quantities have different units. Some people differentiate and say that the two quantities in a ratio have to have a same unit some people don't differentiate and allow "3 dollars per gallon" to be called a ratio as well.

PROPORTION is an equation where two ratios are equal. For example, "3 dollars per gallon" equals "6 dollars per two gallons". Or, 2 teachers per 20 students equals 3 teachers per 30 students. Or,

Of course, for it to be a problem, you need to make one of those four numbers to be an unknown (not given).

Free proportion worksheets
Free worksheets for simple proportion word problems.

How To Solve Mixtures & Addition Based Ratio Problems

Question 10.

Chetan and Shaheen’s salaries are in the ratio 5 : 9. If both of their salaries are raised by Rs. 4200, then the proportion changes to 22 : 27. Find Shaheen’s salary.

A. 9250.95
B. 8058.32
C. 7199.97
D. 13580.45

Solution:

Let Chetan and Shaheen’s salaries be 5x and 9x

Given, 5x + 4200 / 9x + 4200 = 22/27

135x + 113400 = 198x + 92400

Therefore, Shaheen’s salary = 333.33 * 9 + 4200 = 7199.97

Question 11.

Salaries of Preeti and Bina are in the ratio 14 : 15. If both get an increment of Rs. 5300, the new ratio becomes 33 : 35. What is Preeti’s salary?

A. 24000
B. 34980
C. 30100
D. 10200

Solution:

Let Preeti’s salary be 14x, and Bina’s salary be 15x

= 490x + 185500 = 495x + 174900

Therefore, Preeti’s salary = 14 * 2120 + 5300 = 34980

Question 12.

400 g of 25% sugar syrup was added to 600 g of 40% sugar syrup. Find the percentage of the syrup in the mixture.

1.3: Proportions and Rates - Mathematics

SECTION 1.2. RATIOS AND PROPORTIONS

A proportion is similar to a ratio, except that it indicates a part of a whole, and so the numerator arises from the denominator. For instance, a researcher might say that, for every ten students in the residence hall, five were women. Five over ten (5 / 10) is a proportion. Proportions must have all of the numbers in the same units, and are frequently written as fractions.

What happens when you want to write a proportion, but the numbers are given in different units? Suppose you were asked to write the proportion of 3 cups to 56 ounces (3 cups out of 56 ounces). You must write a proportion of either cups to cups (cups:cups) or ounces to ounces (ounces:ounces), so you will have to convert one of the numbers so that both numbers are expressed in the same unit of measurement. Let's convert cups to ounces so we can express the ratio as ounces:ounces. Since there are 8 ounces in one cup, 3 cups are equal to 24 ounces:

3 cups * 8 ounces/cup = 24 ounces

Now the two numbers 3 cups and 56 ounces can be written as the following ratio:

24 ounces to 56 ounces,

24:56,

or, now that the unit of measurement is the same, 24:56 can also be written as a proportion:

You may need to solve some problems involving ratios.

If you divided 36 into two parts in the ratio of 1:2 and one part is a and the other is b, you can find the value of a and b:

You can use these equations to solve for a and b, or you can use the following simple method:

Find out how many units are in 1 part of the ratio. To do this, divide the total by the number of parts.

Number of parts: 1 + 2 = 3.

Number of units in each part: 36/3 = 12.

Then, multiply the number of units in each part by the number of parts in each variable.

a = 1 * 12 = 12

b = 2 * 12 = 24

Percentages are so frequently used that we should spend a little time on them here. The powerful thing about percentages is that they all have the same denominator: 100. When two fractions have the same denominator, comparisions can be made very easily.

Copyright © 2004 by the Regents of the University of Minnesota, an equal opportunity employer and educator.

Common Core Standards for Mathematics

6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, &ldquoThe ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.&rdquo &ldquoFor every vote candidate A received, candidate C received nearly three votes.&rdquo

6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, &ldquoThis recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.&rdquo &ldquoWe paid $75 for 15 hamburgers, which is a rate of$5 per hamburger.&rdquo

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

6.RP.A.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

6.RP.A.3b Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

6.RP.A.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity) solve problems involving finding the whole, given a part and the percent.

6.RP.A.3d Use ratio reasoning to convert measurement units manipulate and transform units appropriately when multiplying or dividing quantities.

Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

7.RP.A.2 Recognize and represent proportional relationships between quantities.

7.RP.A.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

7.RP.A.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

7.RP.A.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

7.RP.A.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

MathHelp.com

The order of the items in a ratio is very important, and must be respected whichever word came first in the ratio (when expressed in words), its number must come first in the ratio. If the expression had been "the ratio of women to men", then the in-words expression would have been " 20 women to 15 men" (or just " 20 to 15 ").

Expressing the ratio of men to women as " 15 to 20 " is expressing the ratio in words. There are two other notations for this " 15 to 20 " ratio:

You should be able to recognize all three notations you will probably be expected to know them, and how to convert between them, on the next test. For example:

There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese as a ratio with a colon, as a fraction (do not reduce), and in words.

They want "the ratio of ducks to geese", so the number for the ducks comes first (or, for the fractional form, on top). So my answer is:

Consider the above park, with 16 ducks and 9 geese. Express the ratio of geese to ducks in all three formats.

This time, they want me to give them "the ratio of geese to ducks". I'll be using the exact same numbers but, in this case, the number of geese comes first (or, for the fractional form, on top). So my answer is:

The numbers were the same in each of the two exercises above, but the order in which they were listed differed, varying according to the order in which the elements of the ratio were expressed. In ratios, order is very important.

Let's return to the 15 men and 20 women in our original group. I had expressed the ratio as a fraction, namely, . This fraction reduces to . This means that we can also express the ratio of men to women in the group as being , 3 : 4 , or " 3 to 4 ".

This points out something important about ratios: the numbers used in the ratio might not be the absolute measured values. The ratio " 15 to 20 " refers to the absolute numbers of men and women, respectively, in the group of thirty-five people. The simplified or reduced ratio " 3 to 4 " tells us only that, for every three men, there are four women. The simplified ratio also tells us that, in any representative set of seven people (3 + 4 = 7) from this group, three will be men. In other words, the men comprise of the people in the group. These relationships and reasoning are what we use to solve many word problems:

In a certain class, the ratio of passing grades to failing grades is 7 to 5 . How many of the 36 students failed the course?

The ratio, " 7 to 5 " (or 7 : 5 or ), tells me that, of every representative group of students, five failed. By "representative group", I mean a group which has the same ratio of students as are in the entire class. I can figure out the size of this group by using the ratio they've given me. The size of the representative group will be the sum of its parts:

So the representative group has 12 students in it, of which 7 passed and 5 failed. In particular, the fraction of the group that failed is given by dividing the number of flunking students by the total number of students in the representative group. That is:

So of the group flunked and, because this group is representative, of the entire class flunked. This means that I can now find the number of students in the entire class that flunked (this exercise is depressing!) by multiplying the fraction from the representative group by the size of the whole class:

So, of the class of 36 students, the class was not passed by:

The ratio from a representative group can also be used to provide percentage information.

In the class above, what percentage of students passed the class? (Round your answer to one decimal place.)

I already know that the representative group contains 12 students, of which 7 passed the class. Converting this to a percentage (by dividing, and then moving the decimal point, as explained here), I get:

They want the answer rounded to one decimal place, so my answer is:

In the park mentioned previously, the ratio of ducks to geese is 16 to 9 . How many of the 300 birds are geese?

The ratio tells me that, of every representative group of 16 + 9 = 25 birds, 9 are geese. That is, of the birds are geese. I can use this fraction from the representative group to find the answer for the entire group:

This is the number they're wanting. In the entire park, there are:

Generally, ratio problems will just be a matter of stating ratios or simplifying them. For instance:

Express the following ratio in simplest form: $10 to$45

This exercise wants me to write the ratio as a reduced fraction. So first I'll form the fraction, and then I'll do the cancelling that leads to "simplest form".

The dollar signs cancelled off, too, because they were the same, top and bottom, in the ratio's fractional form. So my answer is:

This reduced fraction is the ratio's expression in simplest fractional form. The units (being the "dollar" signs) cancelled on the fraction because the units (namely, the " \$ " symbols) were the same on both values.

When both values in a ratio have the same unit or designator, there should be no unit or designator on the reduced form of the ratio. The units aren't factors, exactly, but they'll cancel in the same manner as do factors.

Express the following ratio in simplest form: 240 miles to 8 gallons

The terms in this ratio have different units, so they won't cancel off there will be units on my simplified ratio. My simplification looks like this:

This particular ratio of units, "(miles)/(gallon)", has its own simplified form namely, "miles per gallon", which is abbreviated as "mpg". So, in standard English, my answer is:

In contrast to the answer to the previous exercise, this exercise's answer did need to have units on it, since the units on the two parts of the ratio (namely, the "miles" and the "gallons") were not the same, and thus did not cancel each other off. When a ratio ends up with units (or dimensions) on it, the ratio may also be referred to as a "rate". In the case of the exercise above, the rate was the distance covered per unit-volume of fuel.

What is the length, in feet, of the playing area of an American football field (that is, the length of the field exclusive of the "end" zones)?

I know that the length of an American football field, exclusive of the "end" zones, is 100 yards. I also know that 3 feet are equal to 1 yard. I can set up this equality as a ratio. Because they've given me a measurement in "yards", I'll want the unit of "yards" to cancel off in my multiplication. Because of this, I'll state my ratio (of yards to feet) with the "feet" on top:

Now I can multiply the length they've given me by my conversion factor (being the ratio above), and simplify:

This value is the number they're wanting, so my answer is:

For more on this topic, look at the "Cancelling / Converting Units" lesson.

Ratios are the comparison of one thing to another (miles to gallons, feet to yards, ducks to geese, et cetera). But their true usefulness comes in the setting up and solving of proportions.

You can use the Mathway widget below to practice converting ratios, expressed in "odds" notation, to fractional form. Try the entered exercise, or type in your own exercise. Then click the button and select "Convert to a Simplified Fraction" to compare your answer to Mathway's. (Or skip the widget, and continue on to the next page.)

(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)

Quiz 1: Ratio, Proportions and Formulas

Remember I = E/R, where (I = amps) against resistance (R = ohms) in the electric field (E = volts.).

In first gear, or low gear, an automobile's engine runs about three times as fast as the drive shaft. In second gear, the engine does not have to run as fast usually it runs about 1.6 times faster than the drive shaft. Finally, in third, or high gear, the engine runs at the same speed as the drive shaft.

Transmission in second gear

In first gear, or low gear, an automobile's engine runs about three times as fast as the drive shaft. In second gear, the engine does not have to run as fast usually it runs about 1.6 times faster than the drive shaft. Finally, in third, or high gear, the engine runs at the same speed as the drive shaft.

Transmission in first gear

Portions of beets served = 6 oz.

To calculate the number of pounds of meat required to feed diners do the following: for poultry or meat with bones, divide the number to be served by 2.5 for meat with little or no bone, divide the number to be served by 3.5 and for fish fillets divide the number to be served by 3.5.

Meat served = chicken with a bone

Remember: (0.45)(w-lb) = (w-kg) and the recommended protein in grams is numerically equal to the person's weight in kilograms.